ABSTRACT

In this work the author introduces and studies the construction of the crossed

product of a von Neumann algebra M decomposed into the direct integral M =

Jx M(x)d/j,(x) by an equivalence relation on X with countable cosets.

This construction is the generalization of the construction of the crossed prod-

uct of an abelian von Neumann algebra by an equivalence relation introduced by

J. Feldman and C. C. Moore in [15].

Many properties of this construction are studied. In particular, the structure

theorem generalizing Theorem 1 in [15] is proved. The generalizations of the Spec-

tral Theorem on Bimodules (see [25, Theorem 2.5]) and of the theorem on dilations

(see [26, Theorem 1]) are proved too.

1991 Mathematics Subject Classification. 47D25, 46L10, 47A20.

Key words and phrases. Von Neumann algebra, crossed product, equivalence relation, bimodule,

subalgebra, dilation.

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